Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model

Details Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model

2008-12-29

arxiv.org/abs/0812.4876v1

Journal of Mathematical Analysis and applications 361, 2 (2008) 533-542

Mathematics

Analysis of PDEs

Scientific paper

10.1016/j.jmaa.2009.07.034

The Keller-Segel system describes the collective motion of cells that are attracted by a chemical substance and are able to emit it. In its simplest form, it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. This paper deals with the rate of convergence towards a unique stationary state in self-similar variables, which describes the intermediate asymptotics of the solutions in the original variables. Although it is known that solutions globally exist for any mass less $8\pi $, a smaller mass condition is needed in our approach for proving an exponential rate of convergence in self-similar variables.

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